Theorem rnun 5099

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5096	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4887	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4893	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2227	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4693	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4693	. . 3 |- ran A = dom `' A
7		df-rn 4693	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3329	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2237	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1373   u. cun 3168   `'ccnv 4681   dom cdm 4682   ran crn 4683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3643  df-pr 3644  df-op 3646  df-br 4051  df-opab 4113  df-cnv 4690  df-dm 4692  df-rn 4693

This theorem is referenced by:  imaundi  5103  imaundir  5104  rnpropg  5170  fun  5458  foun  5552  fpr  5778  fprg  5779  sbthlemi6  7078  exmidfodomrlemim  7324